Well, you certainly don't want to be in bad shape....and there will be some who are. I was in the best shape of my life when I went in, actually qualified for the special services (they had a course we all ran and I came in 3rd), but was disqualified due to my depth perception not being up to par.
I'm sure that boot camp is different to some degree for each branch, but if it's at all like what I experienced you'll have plenty of other stuff on your plate to worry about without having to deal with your body being out of shape. We had to guard our doors at night, and had to wake up often, you get up super early as well, so lack of sleep can wear you down. That can be quite hard during the day when you have classes (often after eating). If your eyes close you have to (or at least 20+ years ago) stand in the back of the class for the rest of the time (I avoided that due to not being able to sleep easily my whole life....only time that has been a blessing).
Honestly the first 2 weeks are the "hardest" and it's not that bad if your young and in shape. It wasn't "fun"...and I was tired, but...there was plenty of decent food...and it was actually less dull then most other times in the service (your usually doing SOMETHING).
Unless things have changed radically though, you won't be playing any video games while your there lol.
As far as which game/s from the list you should play, you have both FPS and RPG games on your list...and hybrids. I'd say go with the hybrids if you like both genre's, so go with Mass effect, Dues Ex and if you can Alpha protocol. Neverwinter can take up all your time by itself, so hold off on that, balders gate is old school...and later you can play it on an ipad or something from anywhere...so hold off on that as well. The witcher is hit or miss for alot of people, it's more pure RPG...and can also take a very long time if you really enjoy the series. If you already own it...you could try it and see if you love it....but I'd go with the others first.
Also, based on what you seem to like, if you have an Xbox (which it sounds like you do...based on Mass effect, you might try a game called Too Human. It's a bit of a hybrid RPG/action game....and one of my favorites although not everyone liked it. It is available for about $20 on amazon...not sure i fyou can get it at game stores anymore.
Back to the fitness, I would run every day, and get used to waking up early. Video games may actually be a good way to train yourself to deal with less sleep if your used to sleeping 8 hours. You might consider dance dance revolution or a game like that to get your fitness level up more as well. Not sure of your age etc...but I just played alot of sports when I was young which took care of the problem for me.
I'm not sure how much boot camp has changed but when I went we had 1 person try and commit suicide in our flight, and 2 others bugged out (one said he was a homosexual and I guess described things that where graphic enough that they let him out, the other I think they threw in the CC...been like 30 years). Some people do have a hard time with boot camp, so just be prepared. It's not really hard, so....I think...if you have your head on straight when you go in, are in fairly decent shape, and don't cave under pressure/stress right away normally, you should be fine. After the early part of boot camp and through much of the training you may be LESS happy due to having to wait around so much and do nothing. That was my least favorite part. It may not be so bad anymore with so many portable ways of entertaining yourself, but you may not be allowed to use it. Even at boot camp towards the end I ran as a courier around the base and would literally just have to wait till something needed to be delivered.
Anyway best of luck to you.